Metrix MO 2020 8p

geometry problems from Metrix Mathematical Olympiad 2020 and the Shortlist (unused problems) with aops links

2020

first posted here

collected inside aops here

contest problems

2020 Metrix MO p3

Does there exist a heptagon $P$ satisfying the following:

There does not exist a square with exactly three of the vertices on the boundary of $P$.

 by MNJ2357

2020 Metrix MO p5

$ABC$ be a scalene triangle. Let $D$ be a point on $\overline{BC}$ such that $\overline{AD}$ is the angle bisector of $\angle BAC$. Let $M$ be the midpoint of $\widehat{BAC}$ of the circumcircle of $\odot(ABC)$. Let $\overline{DN}\perp\overline{BC}$ where $N$ lies on $\overline{AM}$. $K$ be a point on segment $NM$ such that $3KM=KN$. Let $P,Q$ be two points on $\overline{AM}$ such that $KP=KQ$ and $BCPQ$ lie on a circle $\omega$. Show that the reflection of $D$ over $A$ lies on $\omega$.

by Amar_04

sample problems

2020 Metrix MO Sample p1

An acute triangle $ABC$ ($AB\neq AC$) with circumcircle $\omega$ is given, and $M$ is the midpoint of arc  BAC . Suppose $D$ is the point on segment $BC$ such that the circumcircle of $MAD$ is tangent to $BC$, and suppose $P$ is the point on segment $BC$ such that $AP$ and $MD$ concur on $\omega$. Let $Q$ be the intersection of $MA$ and $BC$, $N$ be the midpoint of $MD$, and $X$ be the intersection of $NQ$ and $MP$. Prove that $\angle MDB=\angle XDC$.

by MNJ2357

2020 Metrix MO Sample p4

The point $H$ is a orthocenter of the triangle $ABC$ , given the conditions that $\odot(ABH)\cap HC=T,\odot(ACH)\cap HB=K$, $\odot(KTA)\cap \odot(ABC)=N$ , $AH\cap \odot(HTK)=P,\odot(HTK)\cap \odot(PNA)=Q$

$\odot(HBC)\cap HQ=M$ with $HACB'$-cyclic, $HABC'$-cyclic. $\angle BAH=\angle CC'A,\angle CAH=\angle BB'A$

$\odot(ABB')\cap \odot(ACC')=H'$ .Then prove that $H',B',C',M$-cyclic

by Functional_equation

unused problems

2020 Metrix MO Shortlist g1

Let $ABC$ be a triangle with centroid $G$. $GD$, $GE$, $GF$ are symmedians of triangles $GBC$, $GCA$, $GAB$, respectively. $L$ is symmedian point of $ABC$. $L^*$ is inversion of $L$ in circle $(ABC)$. Prove that $AD$, $BE$, $CF$ and $GL^*$ are concurrent.

by buratinogiggle

2020 Metrix MO Shortlist g2

Let the points $H$ and $N_9$ to be the orthocenter and the Nine-Point center of $\vartriangle ABC$ respectively and let $R$ to be the circumradius of $\vartriangle ABC$. If $\angle BAC = \alpha$ , $\angle ABC = \beta$, $\angle ACB = \gamma$. Prove that $$\left( \frac{2HN_9}{R}\right)^2 \le  9 - 8\sqrt3 \sin \alpha \cdot \sin \beta \cdot \sin \gamma.$$

 by Functional_equation

2020 Metrix MO Shortlist g3

Let $D, E,$ and $F$ be the respective feet of the $A, B,$ and $C$ altitudes in $\triangle ABC$, and let $M$ and $N$ be the respecive midpoints of $\overline{AC}$ and $\overline{AB}$. Lines $DF$ and $DE$ intersect the line through $A$ parallel to $BC$ at $X$ and $Y$, respectively. Lines $MX$ and $YN$ intersect at $Z$. Prove that the circumcircles of $\triangle EFZ$ and $\triangle XYZ$ are tangent.

by mathman3880

2020 Metrix MO Shortlist g4

In triangle $ABC$ denote $I_A$ as the center of the excircle wrt $A$. Let the excircle touch the sides $AB, BC, CA$ at $M, N, P$ respectivly. Let $T$ be the midpoint of $MP$ , $TC$ and $MN$ hit at $T'$ and $BC$ , $PM$ hit at $T''$ . Prove $I_A$ is the orthocenter of $TT'T''.$

by Mr.C